Why should you not use even though?

There's nothing wrong with using "even though." I imagine you've had a teacher caution you against starting a sentence with "even though" because you've been writing fragments or incomplete sentences. For example, if I say "Even though I was tired," you'd wonder why I said "even though." I could just say "I was tired." Or I could finish the rest of my thought in a single sentence: "Even though I was tired, I stayed up late to finish my homework."